In Illinois, how do I legally evict my tenant that does not have a lease for non payment of rent?

It doesn't matter if they have a written lease or not. If they pay rent (or are supposed to pay rent) on a monthly basis then they are a month-to-month tenant by law.

You would evict them the same way you would evict a tenant with a written lease. You give them written notice (either 3, 5, 7 or 10 days, depending on your state's law) to pay or vacate. If they fail to pay or vacate within that time then you file a lawsuit for eviction.

Go to the sticky at the top of the Real Estate Law forum. It has links to the landlord/tenant laws for all of the states. You can get the details for your state there.

Do you have the link to the forum? I don't see a real estate law forum that has such pages (pages by state for landlord tenant laws) - Thanks!